a.a. gerasimov, b.a. grigoryev, i.s. aleksandrov

30 Proceedings of gas industry research and development

Vesti gazovoy nauki · Digest 2014

UDC 622.324

A.A. Gerasimov, B.A. Grigoryev, I.S. Aleksandrov

Calculation of Phase Equilibriums of Complex Hydrocarbon Mixes on the Basis of Multi-Constant Generalized State Equations

Phase behavior of multi-component hydrocarbon mixes of natural or man-made origin is modeled, as a rule, using different cubic state equations (SE). Most widely the equations of Peng–Robinson (PR) [1], Soav–Redlikh–Kvong (SRK) [2, 3] and the four-constant equation developed by Brusilovsky (BRS) are used [4]. This list can be continued, though there are other kinds of cubic equations, which are less developed and, as far as the authors know, do not have any signifi cant advantages over the specifi ed SE. At the same time cubic SE have one essential drawback – they do not provide high calculation accuracy of thermodynamic properties (TDP). This matter has been paid the most careful attention of scientifi c literature, including the works of the authors [5–7]. Alternatively, in [7], it is proposed to calculate TDP of complex hydrocarbon mixes on the basis of two generalized multi-constant SE describing all thermodynamic properties of n-alkanes and cyclic hydrocarbons, correspondingly, in the temperature range from triple point to 700 K at pressure up to 100 MPa. In this work, authors propose a method for calculating phase equilibrium of complex hydrocarbon mixes on the basis of specifi ed multi-constant SE.

Modeling composition of a complex hydrocarbon mixSince we are studying hydrocarbon mixes for which individual hydrocarbon composition

is not defi ned, there is a problem of its identifi cation (modeling). As a rule, composition of liquid hydrocarbon mixes is modeled by pseudo-components (sub-fractions). These issues are partially reviewed in [5]. Pseudo-component is characterized by a set of integrated indicators of its composition – average boiling temperature (Tb), average molar weight (М), relative density ( 20

4ρ ), refraction index ( 20dn ), etc. However, in real conditions information

about pseudo-components proves to be rather limited, and calculation methods have to be used [5, 8]. Modeling composition by pseudo-components is carried out on the basis of data about fractional composition, which can be obtained by distillation in (cyrillic) АРН-2 apparatus for oil rectifi cation [9] (the data for the true boiling point curve (TBP) is derived), either using Engler equipment (at that, it is necessary to recalculate temperatures to TBP curve [8]). Distillation curve (fi g. 1) is broken down into fi ve sub-fractions (pseudo-component) corresponding to 10, 30, 50, 70, 90 % distillate yields by volume. If we study a fraction of representative oil or gas condensate, for which, as a rule, distillation in 10-degree fractions on АРН-2 device is carried out with defi ning physical and chemical indicators ( 20dn , 20

4ρ , М) for each fraction, we need to approximate specifi ed data as a function of Tb in all investigated temperature range, and according to obtained relationships to defi ne composition indicators for each sub-fraction. However, most often such data is not available. Therefore, we will consider an option with minimum information about physical and chemical properties (PCP) of the mix – with distillation curve, 20

dn , 204ρ , and M.

Composition identifi cation is carried out using the following algorithm:1) on the basis of distillation curve, we defi ne the Tb,i values for each of fi ve sub-

fractions, corresponding to 10, 30, 50, 70, 90 % distillate yield by volume; 2) then, the general average boiling temperature Tb,V = (T10 + T30 + T50 + T70 + T90) / 5

is defi ned;3) if any of three indicators – 20

dn , 204ρ and M – is unknown, it must be calculated using

empirical formulas recommended in [7];

Keywords:

thermodynamic properties,

phase equilibria, equation of state,

fugacity, hydrocarbon,

oil,gas condensate.

31

Scientific and technical collection book

Actual problems of research of stratal hydrocarbon field systems

4) the number of carbon atoms (NC) to be calculated in complex mix averaged molecule (Tb = Tb,V) is defi ned by formula

2 2

124,3645 33,34334 103,4013ln100 100

100 100 1001349,8 4168,875 4215,153 ,

b bC

b b b

T TN

T T T

⎛ ⎞= − + − −⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞− − +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ (1)

and 204ρ and M incriminated to hydrocarbon with NC and Tb are calculated by formulas

20 4 24

2 3

1,797950,748283 7,790955 10 6,701407 10 ln( )

9,042302 27,61435 ,

CCC

C C

N NN

N N

− −ρ = − ⋅ + ⋅ − +

+ − (2)

2 3

6008,461 25999,71 47376,32939,126 17,4411 256,9713ln( ) .C CC C C

M N NN N N

= + − − + −

(3)

Authors obtained formulas (1) – (3) by approximation of Katz and Firoozabadi data [11], who had published generalized results of PCP studies for gas condensates and de-gassed oil at several dozens of deposits.

Then, correction factors Kρ and KM are defi ned:204

204

,( )C

KNρ

ρ=ρ

(4)

, ( )M

C

MKM N

= (5)

where 204ρ and M – experimental values of indicators; 20

4ρ (NC), M(NC) – values of the same indicators calculated by formulas (2) and (3);

5) with known ibT values for sub-fractions, using formulas (1) – (3) one calculates (

iCN ),

M(iC

N ), and to reduce systematic error – corrects them using formulas

Fig. 1. Distillation curve for Mangyshlak oil fractions at 140–180 °С

140

150

160

170

180

190

0 10 20 30 40 50 60 70 80 90 100

Distillate yield by volume, %



20 204 4 ( ,)i CiN Kρρ = ρ (6)

C( ) .ii MM M N K= (7)

Then, again it is necessary to correct values in relation to density in the assumption of Amagat’s law fulfi lment:

5

20 2014 4

1 , ( )

i

i i

r=

=ρ ρ∑ (8)

then the corrected density value will be defi ned by relationship5

20 20 204 4 4 20

1 4

( ( ))

(

) ,i

i

ii C

i C

rN

N=

ρ = ρ ρρ∑ (9)

where ri – share of volume (ri = 0,2);6) perform recalculation of model mix composition from ri volume shares into xi molar

ones, and also correction of molar weights values of Mi pseudo-components:204

2042054

201 4

,

(

( )

( ))

i i

ii i

i C i

rMxrM N=

ρρ

=ρ

ρ∑

(10)

5

1

( );

( )

C ii

i C ii

M N MM

x M N=

=

∑ (11)

7) using formulas given in [8], defi ne the values of pseudo-critical properties (Трс, ррс, ρрс) and acentricity ω factor for each sub-fraction;

8) at known values of composition integrated indicators 20dn , 20

4ρ and M using n–ρ–M method [10] or, if the value of kinematic viscosity factor (υ) is known, by n–ρ–υ–M method [9], calculate relative content of paraffi n (хР), naphthenic (хN) and arenose (xA) compounds in hydrocarbon mix. Further, these values are identifi ed with relative content of paraffi n, naphthenic and arenose hydrocarbons in the mix (possibility of such identifi cation is shown in [12, 13]). Then relative content of cyclic-structure hydrocarbons is defi ned:

хC = хN + xA; (12)

9) assuming that the parity between paraffi n and cyclic structures is preserved for each sub-fraction, we defi ne the molar share of paraffi n and cyclic hydrocarbons in each sub-fraction:

, where 1,..., . , 5P P C Ci i i ix x x x x x i= = = (13)

Thus, a ten-component model mix is obtained. At that, pseudo-critical properties, acentricity factor and molar weight of paraffi n and cyclic components in sub-fraction are assumed equal to values defi ned for this sub-fraction.

Rules for transition from ‘quasisingle-fluid’ model to ‘ideal-solution’ modelAs shown in [7], ‘quasisingle-fl uid’ model provides high accuracy calculation of all

thermodynamic properties of complex hydrocarbon mixes both in liquid and gas phases, and in supercritical area of state parameters. Therefore, the natural advantages of this model have been preserved, and the model itself has been transformed into the ‘ideal-solution’ mode, which would make it possible to carry out phase equilibrium calculations. To defi ne the rules of such transition, one must set equal the reduced Helmholtz excess energy

33



( , , )r xα δ τ ) expressed in ‘quasisingle-fl uid’ approximation and the form of linear model (ideal solution):

1 1

1 1

5 5

1 1 1 1

exp( )

e

( ) ( )

xp( )

pol

i i i i i

pol

pol

i i i i i

pol

pol

i i i

i i

pol

n NT d T d lP P P

i ii i n

n NT d T d lC C C

i ii i n

n NT d TP P P P

n ij j j n ij j ji j i n j

n n x

n n x

K n x K n x

= = +

= = +

= = = + =

⎡ ⎤τ δ + τ δ −δ⎢ ⎥

⎢ ⎥⎣ ⎦⎡ ⎤

τ δ + τ δ −δ⎢ ⎥⎢ ⎥⎣

+

+ =

=

⎦

⎛ ⎞ ⎛ ⎞ω τ δ + ω τ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

∑ ∑

∑ ∑

∑ ∑ ∑ ∑5 5

1 1 1 1

exp( )

( ) ( ) exp( ) .

i i

pol

i i i i i

i i

pol

d l P

n NT d T d lC C C C C

n ij j j n ij j ji j i n j

x

K n x K n x x= = = + =

⎡ ⎤δ −δ⎢ ⎥

⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞ ⎛ ⎞

ω τ δ + ω τ δ −δ⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

+

+ ∑ ∑ ∑ ∑

(14)

Setting equal relevant summands at similar degrees of reduced temperature (Ti) and reduced density (di), one obtains formulas for defi ning correcting factors

i

PnK and

i

CnK :

5 5

1 1

( ) ( ), ,

( ) ( ) i i

P CP Ci in n

P Cij j j ij j j

j j

n nK K

n x n x= =

ω ω= =

ω ω∑ ∑ (15)

where ( )Pin ω , ( )C

in ω – factors of generalized fundamental state equation (FSE), calculated correspondingly by the known ω values for paraffi n hydrocarbons and cyclic composition hydrocarbons in the mix; ( )P

ij jn ω , ( )Cij jn ω – coeffi cients of generalized FSE calculated by

the known for jth sub-fraction value ωj for paraffi n hydrocarbons and cyclic hydrocarbons, correspondingly. The form generalized FSE, values of coeffi cients and exponents are given in [7].

Reduction parameters for linear model (ideal-solution model) are calculated by relationships

10

,1

( ) ,mr T j pm jj

T x K x T=

= ∑ (16)

10

1 ,

1)

1(

,m jjr pm j

K xx ρ

=

=ρ ρ∑ (17)

where correcting factors are defi ned by relationships

, ,10 10

, , 1 1 , ,

, ,1cm

calc m calc mT

j calc m j jj j calc m j

TK K

x T xρ

= =

ρ= =

ρ∑ ∑ (18)

where ρm, Tm – critical density and temperature of the mix; Tcalc,m,j, ρcalc,m,j – pseudo-critical properties of pseudo-components of the mixes calculated by formulas, recommended in the source [8].

Using this model, beginning boiling and condensation (dew-point) points were calculated, and this data has been compared with experimental data.

Algorithm of phase equilibriums calculationSetting of the problem. At specifi ed temperature (Т) and known molar composition

of initial mix (z ), we need to defi ne pressure at the beginning of boiling (psv) together with equilibrium concentrations in gas phase y, or pressure at the beginning of condensation (psl) together with equilibrium concentrations in liquid phase x .

Method of psv and psl calculation is reduced to the following:• psv together with equilibrium concentrations in gas phase y are defi ned by solving

equations system



, ,ln ( , , ) ln ( , , ) 0, 1, 2, ..., ,i j i vf p T z f p T y i N− − = (19)

1

1 0; N

ii

y=

− =∑ (20)

• psl together with equilibrium concentrations in liquid phase x are defi ned by solving equations system

, ,ln ( , , ) ln ( , , ) 0, 1, 2, ..., ,i v i jf p T z f p T x i N− = = (21)

1

1 0, N

ii

x=

− =∑ (22)

where fi,v, fi,l – volatilities (fugitivities) of the ith component in gas and liquid phases, correspondingly.

Volatilities of components in gas and liquid phases have been calculated on the basis of known thermodynamic identities for generalized FSE (6). Systems of equations (19) – (22) have been solved in iterative process using the Newton–Rafson method; those for linearized systems of equations on each step of iterative process – by the Gauss method. Calculation algorithm, methodical issues and selection of initial approximations are described in works [4, 5].

Experimental data and calculation resultsThough there are works (e.g., [14]), dedicated to calculation of boiling beginning and

condensation beginning points, in literature there are no published experimental data about phase equilibrium of complex hydrocarbon systems which could be generalized or checked. Even in published data about boiling beginning and condensation beginning points, there is no information allowing to carry out solution identifi cation. Hence, authors had to use only the experimental data received in 1980–1990 in the Thermal Physics branch laboratory of Grozny Oil Institute under the guidance of professor Yu.L. Rastorguev and professor B.A. Grigoriev [13, 15–17].

On the basis of obtained relationships, Fortran-based programs were developed allowing to calculate boiling beginning and condensation beginning points and composition of co-existing phases. Phase equilibriums of hydrocarbon fractions investigated in [13, 15–17], and also boiling beginning points for 13 hydrocarbon fractions and condensation beginning points for 7 fractions have been calculated. Calculation has been made on the basis of two generalized fundamental state equations using both linear model, which defi nes reduction parameters by formulas (16) and (17) (FSE LM), and Kunts–Wagner model [18] defi ning reduction parameters by formulas (23) and (24) without taking into account interactions (FSE KW), i.e. parameters of binary interaction βv,ij, βT,ij, γv,ij, γT,ij were assumed equal to ‘one’.

31

2, , 2 1/3 1/3

1 1 1, , , ,

1 1 1 1 12 ,( ) 8

N N Ni j

i i j v ij v iji i j ir m i v ij i j m i m j

x xx x x

x x x

−

= = = +

⎛ ⎞+= + β γ +⎜ ⎟⎜ ⎟ρ ρ β + ρ ρ⎝ ⎠∑ ∑ ∑ (23)

12 0,5

, , , , ,21 1 1 ,

( ) 2 ( ) ,N N N

i jr i m i i j T ij T ij m i m j

i i j i T ij i j

x xT x x T x x T T

x x

−

= = = +

+= + β γ

β +∑ ∑ ∑ (24)

where ρm,i, Tm,i – critical density and temperature of the ith component of the mix.As the cubic state equations are widely used for calculation of phase equilibriums,

authors have also performed calculations by SRK and BRS cubic SE.Nature of deviations for two Mangyshlak oil fractions is shown on fi g. 2, 3. For other

fractions, deviations are of about the same size and nature. Analysis of deviations for all investigated hydrocarbon mixes shows that calculation accuracy of pressures in the beginning of boiling and condensation is about the same for all equations and models. At that, one should pay attention to normally systematic character of variations in results obtained by different equations. It seems that rather high deviations can be explained by cumulative effect of different concomitant factors – inaccuracy of distillation curve, errors in defi nition

35



of physical and chemical properties (especially Tb and M), and, as consequence, errors in identifi cation of pseudo-components, errors of phase transition points experimental defi ning. Methods of phase equilibriums calculation can be made more accurate only at availability of reliable experimental data about TDP and phase behavior of complex modeling mixes of known composition.

As a result, methodology for calculating TDP and phase equilibriums of complex hydrocarbon mixes in temperature range from the beginning

-10

-5

0

5

10

15

20

360 380 400 420 440 460 480 500

Dev

iatio

n, %

Temperature, K

FSE LM FSE KW BRS SRK

-5

0

5

10

15

20

25

30

460 480 500 520 540 560 580 600 620 640

Dev

iatio

n, %

Temperature, K

Fig. 2. Deviations in experimental data of initial boiling pressures from values calculatedby different models and formulas: a – IBT–62 °С fraction of Mangyshlak oil;

b – 140–180 °С fraction of Mangyshlak oil

a

b

of hardening up to 700 K and at pressures up to 100 MPa has been developed. This method is based on two generalized FSE describing with rather high accuracy all thermodynamic properties of n-alkanes and hydrocarbons of cyclic composition, and differs from existing methodologies by higher TDP calculation accuracy and wider ranges of applicable temperatures and pressures.



References1. Peng D.-Y. A new two-constant equation of state /

D.-Y. Peng, D.-Y. Robinson // Ind. Eng. Chem. Fundamen. – 1976. – V. 15. – P. 59–64.

2. Graboski M.S. A modifi ed Soave equation of state for phase equilibrium calculations. 1: Hydrocarbon systems / M.S. Graboski, T.E. Daubert // Ind. Eng. Chem. Process Des. Dev. – 1978. – V. 17. – P. 443–448.

Dev

iatio

n, %

Temperature, K

FSE LM FSE KW BRS BRS

Dev

iatio

n, %

Temperature, K

-40

-30

-20

-10

0

380 400 420 440 460 480 500

-40

-30

-20

-10

0

10

510 530 550 570 590 610 630

Fig. 3. Deviations in experimental data of initial boiling pressures from values calculatedby different models and formulas: a – IBT–62 °С fraction of Mangyshlak oil;

b – 140–180 °С fraction of Mangyshlak oil

a

b

3. Graboski M.S. A modifi ed Soave equation of state for phase equilibrium calculations. 2: Systems containing CO2, H2S, N2 and CO / M.S. Graboski, T.E. Daubert // Ind. Eng. Chem. Process Des. Dev. – 1979. – V. 18. – P. 300–306.

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Work is executed at fi nancial support of the Russian Federal Property Fund, grant 12-08-00284а.

37



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a.a. gerasimov, b.a. grigoryev, i.s. aleksandrov

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